Introduction Prepared by Sa’diyya Hendrickson

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Package Summary

• The Base – 10 Number System

• Reviewing Long Division

• Understanding Fractions

• Equivalent Fractions

• Examples of Equivalent Fractions

• Improper Fractions and Mixed Numbers

• Ratios and Proportions

• Let’s Play! (Exercises)

High School Math Prep 1 of 10 c© Sa’diyya Hendrickson

Base – 10 System Level: Q

1. numeration system: A system that uses symbols or numerals, to represent num-

bers. e.g. Tally Numeration System, Roman Numeration System, or the Babylonian

Numeration System

2. place value: Place values are used by some numeration systems to give a specific

numerical value to a digit/symbol, depending on its position.

3. Hindu-Arabic Numeration System (Base – 10 system): The Base – 10 sys-

tem is the number system that we presently use, and has the following characteristics:

• The digits used include: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

• Numbers are constructed by grouping sets of 10 so that the place values are

based on powers of 10. When reading from right to left, if a digit has the first

place value, it represents a group of less than ten ones. If a digit has the second

place value, it represents groups of 10 = 101 ones. If a digit has the third place

value, it represents groups of 100 = 102 ones, and so on.

• Place values for integers include: ones place, tens place, hundreds place, thou-

sands place, ten-thousands place, etc. (when reading digits from right to left).

For example, the underlined digit in 38, 157 is in the place.

4. expanded form of a number: The expanded form of a number expresses the

number as a sum of its digits multiplied by their respective place values. For example,

245, 036 = 2(105) + 4(104) + 5(103) + 0(102) + 3(101) + 6(1)

= 200, 000 + 40, 000 + 5, 000 + 30 + 6

Any partially expanded forms will be referred to as semi-expanded.

High School Math Prep 2 of 10 c© Sa’diyya Hendrickson

Long Division Level: Q

Consider the following example: Evaluate 2683÷ 3.

Solution: Let’s begin by considering a semi-expanded form of 2683. Then, using this

form, we will systematically determine how many multiples of 3 we have:

2683 = 26 00 + 83

= (2400 + 200) + 83

= 3(800) + 283

= 3(800) + 28 0 + 3

= 3(800) + (270 + 10) + 3

= 3(800) + 3(90) + 13

= 3(800) + 3(90) + (12 + 1)

= 3(800) + 3(90) + 3(4) + 1

= 3(894) + 1

Therefore 2683÷ 3 = 894 plus a remainder of 1.

This solution helps to illustrate why the method of long division always works! Let’s

take a look at this exercise using that approach. Can you identify the similarities?

High School Math Prep 3 of 10 c© Sa’diyya Hendrickson

Understanding Fractions Level: Q

• Rational Numbers (fractions) (Q): The set of rational numbers is the set Qwhere numbers are of the following form:

a

bwhere a and b are integers and b 6= 0.

Here, a is called the numerator and b is called the denominator. For example, 13

is a rational number with a numerator of a = 1 and a denominator of b = 3.

I. Fractions are division problems! i.e. ab

= a ÷ b. For example, 2683 ÷ 3 = 26833

.

How many ways can we express the number 1 as a fraction?

II. Fractions are symbols that represent a part-to-whole relationship. ab

means

that “a” objects are collected for each group of “b” objects. i.e. The numerator rep-

resents the part and the denominator represents the group that produces one whole.

Below, the shaded regions in each example represent the same relationship: For each

group of five objects, objects are collected.

III. Fractions can be interpreted numerically, where the quantity represents

the part-to-whole relationship. e.g. 25

= 0.4 represents the relationship above.

High School Math Prep 4 of 10 c© Sa’diyya Hendrickson

Equivalent Fractions Level: Q

• equivalent fractions: We say that two fractions are equivalent when they repre-

sent the same part-to-whole relationship.

1. Fill in the shaded region so that the fractions are equivalent:

S1 Compare the denominators by first identifying the number of parts

that make one whole/group from the completed fraction.

Notice that the denominators increased from one group of 4 (on the left side) to

groups of 4 (on the right side).

S2 Identify the part-to-whole relationship by looking at the shaded dia-

gram and complete fraction. Then, shade the incomplete diagram in

a way that maintains this part-to-whole relationship.

Here, for one group of 4, we shade parts. Since we have three groups of

4 on the right, we must shade 3× slices in the diagram.

2. Suppose you get a sales job where you receive $3 for every five-dollar sale made.

(a) If you make a total of six five-dollar sales, how much money was made in all?

(b) How much of this money is your employer obligated to give you in return?

(c) What are the equivalent fractions that represent the part-to-whole relationship

of your pay?

High School Math Prep 5 of 10 c© Sa’diyya Hendrickson

Examples of Equivalent Fractions Level: Q

(a)2

5=

4

10= 0.4

From the diagrams in (II) on page 4, we could see that these fractions were equivalent.

This is also evident using fraction number lines.

(b)1

3=

2

6=

4

12

Note: The only difference in the diagrams below is the number of groups/“wholes”

present. More specifically, the number of groups increases from one group of 3 to two

groups of 3, and then to four groups of 3. Therefore, in each case, we simply continue to

shade one rectangle for each group of three that we see.

High School Math Prep 6 of 10 c© Sa’diyya Hendrickson

Improper and Mixed Level: Q

1. improper fraction: We say that a

fraction is improper when the nu-

merator is greater than the denomina-

tor. For example, 26833

is an improper

fraction and simply means: “2683 di-

vided by 3” or that ”we have a total of

2,683 parts whenever 3 parts produce

one whole.”

2. mixed number: A mixed number has the form of an integer beside a fraction and

represents a number separated into two parts: the total number of wholes, plus the

remaining fractional part. For example, let’s express 26833

as a mixed number.

Solution: From our long division on page 3 we found that 2683÷3 = 894 R1. Therefore,

there are 894 groups of 3 (i.e. 894 wholes), plus ”1 out of 3” remaining. Therefore,26833

= 89413.

1. To create a mixed number from an improper fraction, we simply need to

divide (as show in the example above).

2. Let’s explain how to create an improper fraction from a mixed number.

Well, an improper fraction simply tells us “how many parts we have in all (which

is given by the numerator), whenever a whole consists of a certain number of parts

(which is given by the denominator).” This understanding gives the following result:

Why? Because if a represents the number of wholes, and c represents the number

of parts in each whole, then (a · c) gives the total number of parts within “a” wholes.

Then, we need to add the remaining b parts to get the total!

High School Math Prep 7 of 10 c© Sa’diyya Hendrickson

Ratios and Proportions Level: Q

1. ratio: A ratio is a pair of numbers a and b, expressed in the form a : b where b 6= 0,

and can represent one of the following relationships: part-to-part relationships,

part-to-whole relationships or whole-to-part relationships. Fractions are

commonly used to represent ratios! In general, a : b = ab.

For example, we can use ratios to compare the size of centimetres to metres:

100 : 1. This reads, ”one-hundred to one” since we get one-hundred centimetres for

every metre. This is an example of a part-to-whole relationship, which produces

the fraction: 1001

.

2. equivalent ratio: Two ratios are said to be equivalent if the fractions that repre-

sent them are equivalent.

3. proportion: A proportion is a statement that equates two ratios or fractions.

Suppose you are making pancakes for yourself and two friends. If the recipe

that you are using serves 12 people and calls for 4 eggs, how many eggs will

you need to modify the recipe so that there are no leftovers?

Solution: We can use equivalent fractions or ratios. Using ratios: we know that the

recipe’s ratio of people to eggs is given by 12 : 4. If we are serving 3 people, we want to

satisfy the following proportion: 12 : 4 = 3 : . The strategies are as follows:

High School Math Prep 8 of 10 c© Sa’diyya Hendrickson

Let’s Play! Level: Q

1. Determine the place value of the underlined digit:

(a) 567, 104 (b) 40, 300 (c) 39, 560

(d) 3, 485, 098 (e) 348, 502, 847 (f) 2, 385, 769, 573

2. Use powers of 10 to express the numbers in question 1 in expanded form.

3. Perform long division and express the solutions as mixed numbers.

(a) 2354

(b) 45876

(c) 857657

(d) 1172578

4. Determine an equivalent fraction for each part, Then use the graph paper

(on the next page) to represent these equivalent fractions with diagrams

and number lines:

(a) 27

(b) 35

(c)58

(d) 310

5. Determine the equivalent improper fraction for each mixed number:

(a) 437

(b) 918

(c) 1145

(d) 41112

6. Determine which statements represent proportions. Justify your answer

with an explanation:

(a) 28

?= 8

32(b) 3

7

?= 6

21

(c) 18 : 9?= 2 : 1 (d) 24 : 6

?= 8 : 3

7. Express ratios for the (a) part-to-part (b) part-to-whole and (c) whole-

to-part relationships of the following:

e.g. 4 dogs, 7 cats, and the total number of pets

Solution: (a) 4 : 7 (dogs to cats) or 7 : 4 (cats to dogs)

(b) 4 : 11 (dogs to pets) or 7 : 11 (cats to pets)

(c) 11 : 4 (pets to dogs) or 11 : 7 (pets to cats)

i. 5 red marbles, 12 blue marbles, total number of marbles

ii. 3 cups of green paint, 6 cups of purple paint, total amount of paint in the mixture

High School Math Prep 9 of 10 c© Sa’diyya Hendrickson

Let’s Play! Level: Q

High School Math Prep 10 of 10 c© Sa’diyya Hendrickson